∞\infty-categories

Idea

Generalising how in an ordinary category, one has morphisms going between objects, and in a 2-category, one has both morphisms (or 1-morphisms or 1-cells) between objects and 2-morphisms (or 2-cells) going between 1-morphisms, in an ∞\infty-category, there are k-morphisms going between (k−1)(k-1)-morphisms for all k=1,2,…k = 1, 2, \ldots. (The 00-morphisms are the objects of the ∞\infty-category.)

This is hence a much more encompassing notion of ∞\infty-category than that of (∞,1)-category. It is also much harder to formalize. While there is by now a very good (∞,1)-category theory/homotopy theory of (∞,n)-categories for all n∈ℕn \in \mathbb{N}, the limiting case where n→∞n \to \infty is currently still poorly understood. While there are several existing proposed definitions for what a single ω-category is, in the most general sense, there is no real understanding of the correct morphisms between them, hence of the correct (∞,1)-category of ω-categories. But this may of course change with time.

If all the jj-morphisms in an ∞\infty-category are equivalences in some suitable sense, we call the ∞\infty-category an ∞-groupoid. In this case we can think of the jj-morphisms for j≥1j\ge 1 as “homotopies” and the ∞\infty-groupoid as a model for a homotopy type. By analogy, we can, if we wish, think of an arbitrary ∞\infty-category as a combinatorial model for a directed homotopy type.

There are many different definitions realizing the general idea of ∞\infty-category. Models for ∞\infty-categories usually fall into two classes:

One of the tasks of higher category theory is to relate and organize all these different models to a coherent general theory.

Strict versus weak

There are many different definitions of ∞\infty-categories, which may differ in particular in the degree to which certain structural identities are required to hold as equations or allowed to hold up to higher morphisms.